Let `f(x)={x+1,x >0, 2-x ,xlt=0` and `g(x)={x+3,x<1,x^2-2x-2,1lt=x<2, x-5,xgeq2` Find the LHL and RHL of `g(f(x))` at `x=0` and, hence, fin `lim_(x->0)g(f(x)).`

Let f(x)={x+1,x >0 \n 2-x ,xlt=0 and g(x)={x+3,x 0)g(f(x)).

Let f(x)={{:(x+1", "xlgt0),(2-x", "xle0):}"and"g(x)={{:(x+3", "xlt1),(x^(2)-2x-2", "1lexlt2),(x-5", "xge2):} Find the LHL and RHL of g(f(x)) at x=0 and, hence, find lim_(xto0) g(f(x)).

f(x)={:{(x+1,x<0 , x^2,xlt=0):} , x≥0  and g(x)={x^3, x<1 2x−1, x≥1 find f(g(x)) and its domain and range

Let f (x)= [{:(x+1,,, x lt0),((x-1),,, x ge0):}and g (x)=[{:(x+1,,, x lt 0),((x-1)^(2),,, x ge0):} then the number of points where g (f(x)) is not differentiable.

Let f (x)=(x+1) (x+2) (x+3)…..(x+100) and g (x) =f (x) f''(x) -f'(x) ^(2). Let n be the numbers of real roots of g(x) =0, then:

Let f(x)={{:(x sin.(1)/(x)",",x ne0),(0",",x=0):}} and g(x)={{:(x^(2)sin.(1)/(x)",", x ne 0),(0",", x=0):}} Discuss the graph for f(x) and g(x), and evaluate the continuity and differentiabilityof f(x) and g(x).

If f(x)=sgn(x)={(|x|)/x,x!=0, 0, x=0 and g(x)=f(f(x)), then at x=0,g(x) is

Let g: R -> R be a differentiable function with g(0) = 0,,g'(1)!=0 .Let f(x)={x/|x|g(x), 0 !=0 and 0,x=0 and h(x)=e^(|x|) for all x in R . Let (foh)(x) denote f(h(x)) and (hof)(x) denote h(f(x)) . Then which of the following is (are) true? A. f is differentiable at x = 0 B. h is differentiable at x = 0 C. f o h is differentiable at x = 0 D. h o f is differentiable at x = 0

If f(x)={{:((x)/(sinx)",",x gt0),(2-x",",xle0):}andg(x)={{:(x+3",",xlt1),(x^(2)-2x-2",",1lexlt2),(x-5",",xge2):} Then the value of lim_(xrarr0) g(f(x))

Let f(x)={(2x+a",",x ge -1),(bx^(2)+3",",x lt -1):} and g(x)={(x+4",",0 le x le 4),(-3x-2",",-2 lt x lt 0):} If a=2 and b=3, then the range of g(f(x)) is